The word 'implies' is definitely ambiguous. Saying that "it just means if A is true, then B is true" does not explain anything. John Corcoran wrote a paper ("The Meanings of Implication" Dialogos 25, pp 59-76, 1973) describing 13 different senses of implication without claiming that this is an exhaustive list.
Usually in ordinary language when we say A implies B we are expressing the thought that B follows from A in some way, Implication between A and B expresses some kind of relation and there are many different relations that might qualify. For example, if A is true and B is true, does it follow that "if A then B" is true? It depends. What kind of relation are we trying to express? If it is just what logicians call material, then "if A then B" is true whenever A and B are both true. If we mean that A entails B then not necessarily: we would need some kind of connection between A and B that acts as a guarantee that B always holds whenever A holds. That guarantee might be a syntactic derivability relation, or a semantic or conceptual relation. Or we might have some modal connection in mind. There might be an unvoiced, "necessarily, if A then B".
These differences were known to the stoic philosophers. Philo described a conditional that is the same as what we now call the material conditional: it is not the case that A is true and B is false. Chrysippus described something more like what we now call a strict conditional: it is impossible for A to be true and B false. Some medieval logicians used the material conditional without explicitly describing it. Frege used it in his logic because it solved a problem in expressing universally quantified expressions. It allowed "All S is P" to be represented as "for any x, Sx → Px" where → is the material conditional. Frege did not claim that all conditionals are material, only that it is a connective that is useful for certain purposes.
Russell gave this connective the name material implication, which is a pity, because calling it implication is potentially misleading. It is simply a conditional within the object language. Many people have pointed out that it is better to call it the material conditional. Quine even called it a use/mention error to describe the material conditional as an implication.
The material conditional serves as a conditional under the assumption that "if A then B" is a bivalent, dyadic, truth function. To say it is bivalent means that "if A then B" has a truth value that is always either true or false. To say it is dyadic means that its truth depends only on A and B, not on any third parameter, e.g. some background or context. To say it is a truth function means that its truth value depends only on the truth values of A and B and not on any other properties of A and B. Under these assumptions it is easy to prove that the material conditional is a conditonal. I did so in my answer to this question.
So the material conditional is a conditional. It is a very crude approximation to the meaning of 'if' in ordinary language. but it is extremely useful in formal logic. Logicians wouldn't use it if it weren't. The material conditional plays an essential role within classical logic. However, there are many ways in which conditionals do not behave like the material conditional. I listed some of them in my answer to this question.
Because of the complexity of understanding conditionals, there has been a great deal of study of their logic. It is a complex subject with an enormous literature. The curious thing is that mathematicians have made little contribution to our knowledge of conditionals in the last 50 years. I would say that the serious study of conditionals began roughly in the late 1960s. Since then there have been thousands of published papers and scores of books on conditionals. But this literature has come from (1) logicians within the philosophy community, (2) linguists, (3) cognitive psychologists, and (4) AI researchers working on knowledge representation. Hardly any of it has been contributed by mathematicians. For whatever reason, mathematicians are content to use the material conditional and are usually not even aware of how huge a subject the logic of conditionals is.
To amplify the paragraph that you quoted from another answer of mine...
You seem to be falling into a common mistake of confusing a conditional within the object language with the logical consequence relation. The material conditional is a connective within the object language. It is a truth function, like conjunction and disjunction. It is often represented by any of the symbols ⊃ → ⇒. In the same way that A ∧ B is true under some interpretations and false under others, A → B is true under some interpretations and false under others. You plug in the truth values for A, B and this determines the truth value for A → B.
Logical consequence, also called entailment, is a metalevel relation between a set of sentences Γ and a sentence A. It is often written as Γ ⊨ A to mean that Γ has the semantic consequence A, and Γ ⊢ A to mean that A is formally deducible from Γ. If Γ ⊨ A holds then under any interpretation in which the sentences within Γ are all true, A is also true under that interpretation. Sometimes it is also handy to have a metametalevel consequence relation.
Unfortunately, the word ‘implies’ is ambiguous between these meanings. In particular, mathematicians are often taught to call the material conditional material implication and to read A → B as "A implies B". This is potentially misleading, because it leads to confusing the connective with the entailment relation. What makes the confusion even worse is that many mathematicians use the symbol ⇒ without being clear whether it is supposed to be the material conditional or entailment or some kind of metalevel consequence relation. It doesn’t help that different textbooks use ⇒ differently. For example, Mendelson uses ⇒ for the material conditional, but Enderton uses → for the material conditional and ⇒ for a metalevel consequence relation. Some texts use ⇒ to indicate a sequent. It's quite a mess.
In case you think I am the only one complaining about this, let me quote you a passage from Peter Smith’s book, Introduction to Formal Logic.
There is an unfortunate practice that - as we said - goes back to
Russell of talking, not of the 'material conditional', but of
'material implication', and reading something of the form (α → γ) as α
implies γ. If we also read α ⊨ γ as α (tauto)logically implies γ, this makes it sound as if an instance of (α → γ) is the same sort of
claim as an instance of α ⊨ γ, only the first is a weaker version of
the second. But, as we have just emphasized, these are in fact claims
of a quite different status, one in the object language, one in the
metalanguage. Talk of 'implication' can blur the very important
distinction. (Even worse, you will often find the symbol ⇒ being used
in informal discussions so that α ⇒ γ means either α → γ or α ⊨ γ, and
you have to guess from context which is intended. Never follow this
bad practice!)
To learn about conditionals, a good start is to read some of the articles in the Stanford encyclopedia:
https://plato.stanford.edu/entries/conditionals/
https://plato.stanford.edu/entries/counterfactuals/
https://plato.stanford.edu/entries/logic-conditionals/
Some other references are:
- Jonathan Bennett, Conditionals: A Philosophical Guide (2003).
- David Sanford, If P then Q (2003).
- Ernest Adams, The Logic of Conditionals (1975).
- David Lewis, Counterfactuals (1973).
- Robert Stalnaker, several papers collected in Knowledge and Conditionals (2019).
- Dorothy Edgington, “On Conditionals”, Mind, Vol. 104, pp. 235–329, (1995).
- Angelika Kratzer, Modals and Conditionals (2012).
- Nicholas Rescher, Conditionals (2007).
- Michael Woods, Conditionals (1997).
- Igor Douven, The Epistemology of Indicative Conditionals (2015).
